A constant force `vecF=2hati-hatj-hatk` moves a particle from position vector
`vecr_(1)=-5hati-4hatj+hatk` to position vector `vec r_(2)=-7hati-2hatj+2hatk` . What will be the work done by this force ?

Answer

To find the work done by the constant force \( \vec{F} = 2 \hat{i} - \hat{j} - \hat{k} \) when moving a particle from position vector \( \vec{r}_1 = -5 \hat{i} - 4 \hat{j} + \hat{k} \) to position vector \( \vec{r}_2 = -7 \hat{i} - 2 \hat{j} + 2 \hat{k} \), we can follow these steps:

### Step 1: Calculate the Displacement Vector
The displacement vector \( \vec{S} \) can be calculated as:
\[ \vec{S} = \vec{r}_2 - \vec{r}_1 \]
Substituting the given position vectors:
\[ \vec{S} = (-7 \hat{i} - 2 \hat{j} + 2 \hat{k}) - (-5 \hat{i} - 4 \hat{j} + \hat{k}) \]
\[ \vec{S} = (-7 + 5) \hat{i} + (-2 + 4) \hat{j} + (2 - 1) \hat{k} \]
\[ \vec{S} = -2 \hat{i} + 2 \hat{j} + 1 \hat{k} \]

### Step 2: Calculate the Work Done
The work done \( W \) by the force is given by the dot product of the force vector \( \vec{F} \) and the displacement vector \( \vec{S} \):
\[ W = \vec{F} \cdot \vec{S} \]
Substituting the values of \( \vec{F} \) and \( \vec{S} \):
\[ W = (2 \hat{i} - \hat{j} - \hat{k}) \cdot (-2 \hat{i} + 2 \hat{j} + 1 \hat{k}) \]
Calculating the dot product:
\[ W = (2)(-2) + (-1)(2) + (-1)(1) \]
\[ W = -4 - 2 - 1 \]
\[ W = -7 \text{ Joules} \]

### Conclusion
The work done by the force as it moves the particle from position \( \vec{r}_1 \) to \( \vec{r}_2 \) is \( -7 \) Joules.

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